| L(s) = 1 | − 996·9-s − 1.90e4·17-s + 6.23e4·25-s − 1.17e5·41-s + 2.78e5·49-s − 1.15e6·73-s − 3.18e5·81-s − 1.24e6·89-s − 5.82e6·97-s − 2.40e6·113-s − 5.24e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.89e7·153-s + 157-s + 163-s + 167-s + 1.50e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.36·9-s − 3.88·17-s + 3.98·25-s − 1.70·41-s + 2.36·49-s − 2.96·73-s − 0.600·81-s − 1.76·89-s − 6.38·97-s − 1.66·113-s − 2.95·121-s + 5.30·153-s + 3.10·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3819980141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3819980141\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 166 p T^{2} + p^{12} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 1246 p^{2} T^{2} + p^{12} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 139298 T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2620562 T^{2} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 7504462 T^{2} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4766 T + p^{6} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 37404722 T^{2} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 186397538 T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 539498638 T^{2} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 20219522 T^{2} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 5127476782 T^{2} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 29362 T + p^{6} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 12179022098 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 21501276098 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 7136056942 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 78211344722 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 102921545998 T^{2} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 25565876978 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 27079768798 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 288626 T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 389636558402 T^{2} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 612202422578 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 310738 T + p^{6} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1457086 T + p^{6} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48510611002830480549461972479, −7.34967332654867074688619999081, −7.05418056219885820227570753765, −6.65426891328045881125399581831, −6.62456918085163206003103698366, −6.58087770545728340164780112310, −6.23343770058290382328234963823, −5.51348799327458549648076510403, −5.49026403219388401946602173743, −5.33231755980442617460130377508, −4.95848086052090428345746702824, −4.52006658536806485683922633665, −4.31915632814042672422214760349, −4.15997280836454079389441085318, −3.86789901737189490276555719663, −3.22324946656302809175974682900, −2.77815953032453011642886814292, −2.71946054462999089218796428537, −2.63253947167963744735225251800, −2.25561635713701872014484308518, −1.57102676378991221941045780710, −1.28017627768555605728699843145, −1.12419677699335392670972983746, −0.26946140116293416852650998299, −0.15865540627602380539157107630,
0.15865540627602380539157107630, 0.26946140116293416852650998299, 1.12419677699335392670972983746, 1.28017627768555605728699843145, 1.57102676378991221941045780710, 2.25561635713701872014484308518, 2.63253947167963744735225251800, 2.71946054462999089218796428537, 2.77815953032453011642886814292, 3.22324946656302809175974682900, 3.86789901737189490276555719663, 4.15997280836454079389441085318, 4.31915632814042672422214760349, 4.52006658536806485683922633665, 4.95848086052090428345746702824, 5.33231755980442617460130377508, 5.49026403219388401946602173743, 5.51348799327458549648076510403, 6.23343770058290382328234963823, 6.58087770545728340164780112310, 6.62456918085163206003103698366, 6.65426891328045881125399581831, 7.05418056219885820227570753765, 7.34967332654867074688619999081, 7.48510611002830480549461972479
